throw a party開いて is the te-form of 開く, used here to mean “hold” or “throw” an event. It often connects to what happens next, as in パーティーを開いてくれる (throw a party for someone).
JA: 友達のためにパーティーを開いて、みんなでお祝いした。
EN: We threw a party for our friend and celebrated together.